We are given data for four different training methods, including the sample mean ($\bar{x}_i$) and sample size ($n_i$) for each method. We also know the between-groups sum of squares ($SS_B$) and the total sum of squares ($SS_T$). We want to test for differences among the means of the four training methods using ANOVA.

1. Problem Description

We are given data for four different training methods, including the sample mean ($\bar{x}_i$) and sample size ($n_i$) for each method. We also know the between-groups sum of squares ($SS_B$) and the total sum of squares ($SS_T$). We want to test for differences among the means of the four training methods using ANOVA.

2. Solution Steps

Step 1: Calculate the within-groups sum of squares ($SS_W$).

We know that $SS_T = SS_B + SS_W$, so $SS_W = SS_T - SS_B$.

$SS_W = 46840.64 - 229.33 = 46611.31$

Step 2: Calculate the degrees of freedom.

The degrees of freedom between groups is $df_B = k - 1$, where $k$ is the number of groups. In this case, $k = 4$, so $df_B = 4 - 1 = 3$.

The degrees of freedom within groups is $df_W = N - k$, where $N$ is the total sample size.

$N = n_1 + n_2 + n_3 + n_4 = 35 + 60 + 55 + 20 = 170$

So, $df_W = 170 - 4 = 166$.

The total degrees of freedom is $df_T = N - 1 = 170 - 1 = 169$. We can check this using $df_T = df_B + df_W = 3 + 166 = 169$.

Step 3: Calculate the mean squares.

The mean square between groups is $MS_B = \frac{SS_B}{df_B} = \frac{229.33}{3} = 76.4433$

The mean square within groups is $MS_W = \frac{SS_W}{df_W} = \frac{46611.31}{166} = 280.7910$

Step 4: Calculate the F-statistic.

The F-statistic is $F = \frac{MS_B}{MS_W} = \frac{76.4433}{280.7910} = 0.2722$

Step 5: Conclusion.

The F-statistic is 0.2722 with degrees of freedom 3 and

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6. We would compare this to an F-distribution with 3 and 166 degrees of freedom to find the p-value. Without knowing the significance level (alpha), we cannot make a definitive conclusion to reject or fail to reject the null hypothesis that all means are equal. However, because the F-statistic is substantially less than 1, it's highly probable we fail to reject the null hypothesis, suggesting that there is no statistically significant difference among the means.

3. Final Answer

F-statistic = 0.2722

Degrees of freedom between groups = 3

Degrees of freedom within groups = 166